On Optimal Performance for Linear Time-Varying Systems

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1 On Optimal Performance for Linear Time-Varying Systems Seddik M. Djouadi and Charalambos D. Charalambous Abstract In this paper we consider the optimal disturbance attenuation problem and robustness for linear time-varying (LTV) systems. This problem corresponds to the standard optimal H problem for LTI systems. The problem is analyzed in the context of nest algebra of causal and bounded linear operators. In particular, using operator inner-outer factorization it is shown that the optimal disturbance attenuation problem reduces to a shortest distance minimization between a certain operator to the nest algebra in question. Banach space duality theory is then used to characterize optimal time-varying controllers. Alignment conditions in the dual are derived under certain conditions, and the optimum is shown to satisfy an allpass condition for LTV systems, therefore, generalizing a similar concept known to hold for LTI systems. The optimum is also shown to be equal to the norm of a time-varying Hankel operator analogous to the Hankel operator, which solves the optimal standard H problem. Duality theory leads to a pair of finite dimensional convex optimizations which approach the true optimum from both directions not only producing estimates within desired tolerances, but also allow the computation of optimal timevarying controllers. DEFINITIONS AND NOTATION B(E, F ) denotes the space of bounded linear operators from a Banach space E to a Banach space F, endowed with the operator norm A := sup Ax, A B(E, F ) x E, x 1 l 2 denotes the usual Hilbert space of square summable sequences with the standard norm x 2 2 := x j 2, x := ( x 0, x 1, x 2, ) l 2 j=0 Z k denotes the k step ahead shift operator for some integer k. P k the usual truncation operator for some integer k, which sets all outputs after time k to zero. An operator A B(E, F ) is said to be causal if it satisfies the operator equation: P k AP k = P k A, k positive integers (1) and stricly causal if it satisfies P k+1 AP k = P k+1 A,, k positive integers (2) The subscripts sc, c and the superscript t i denote the restriction of a subspace of operators to its intersection with S.M. Djouadi is with the Faculty of Electrical & Computer Engineering Department, University of Tennessee, Knoxville, TN djouadi@ece.utk.edu C.D. Charalambous is with the Faculty of the Electrical & Computer Engineering Department, University of Cyprus, Nicosia, 1678, chadcha@ucy.ac.cy causal, strictly causal (see [12] for the definition), and timeinvariant operators respectively. stands for the adjoint of an operator or the dual space of a Banach space depending on the context. denotes convergence in the weak topology [7], [10]. I. INTRODUCTION There have been numerous attempts in the literature to generalize ideas about H control theory to time-varying systems. In [15], [16] and more recently [14] the authors studied the optimal weighted sensitivity minimization problem, the two-block problem, and the model-matching problem for LTV systems using inner-outer factorization for positive operators. They obtained abstract solutions involving the computation of norms of certain operators, which is quit difficult since these are infinite dimensional problems. Moreover, no indication on how to compute optimal LTV controllers is provided. In [9] both the sensitivity minimization problem in the presence of plant uncertainty, and robust stability for LTV systems in the l induced norm is considered. However, their methods could not be extended to the case of systems operating on finite energy signals. Analysis of time-varying control strategies for optimal disturbance rejection for known time-invariant plants has been studied in [19], [2]. A robust version of these problems were considered in [18], [8] in different induced norm topologies. All these references showed that for time-invariant nominal plants, time-varying control laws offer no advantage over time-invariant ones. In this paper, the focus is on optimal disturbance rejection for time-varying systems. Using inner-outer factorizations as defined in [3], [13] with respect of the nest algebra of lower triangular (causal) bounded linear operators defined on l 2 we show that the problem reduces to a distance minimization between a special operator and the nest algebra. The inner-outer factorization used here holds under weaker assumptions than [15], [16], and in fact, as pointed in [3], is different from the factorization for positive operators used there. These definitions parallel well known factorizations of L functions w.r.t. H as an algebra on the unit circle. Then the duality structure of the problem showing existence of optimal LTV controllers, and predual formulation are provided. Alignment conditions in the dual of the Banach space B(l 2, l 2 ) are derived under certain conditions. Next, the optimum is shown to satisfy an allpass condition for LTV systems, therefore, generalizing a similar concept known to hold for LTI systems for the optimal standard H problem [21], [22], [24]. The optimum is also shown to be equal to the norm of a time-varying Hankel operator analogous to the Hankel

2 operator, well known in the optimal standard H problem. Duality theory leads to a pair of dual finite dimensional convex optimizations which approach the real optimal disturbance rejection performance from both directions not only producing estimates within desired tolerances, but also allow the computation of optimal time-varying controllers. Our approach is purely input-output and does not use any state space realization, therefore the results derived here apply to infinite dimensional LTV systems. Although the theory is developed for causal stable system, it can be extended in a straighfoward fashion to the unstable case using coprime factorization techniques for LTV systems discussed in [12], [16], [13]. The framework developed here can also be applied to other performance indexes, such as the two-block mixed sensitivity problem considered in [16], or even the optimal robust disturbance attenuation problem considered in [11], [20], [4], [5], [6]. II. PROBLEM FORMULATION In this paper we consider the problem of optimizing performance for causal linear time varying systems. The standard block diagram for the optimal disturbance attenuation problem that is considered here is represented in Fig. 1, where u represents the control inputs, y the measured outputs, z is the controlled output, w the exogenous perturbations. P denotes a causal stable linear time varying plant, and K denotes a time varying controller. z y Fig. 1. P K Block Diagram for Disturbance Attenuation The closed-loop transmission from w to z is denoted by T zw. Using the standard Youla parametrization of all stabilizing controllers [12] the closed loop operator T zw w u can be written as [2], [19] T zw = T 1 T 2 QT 3 (3) where T 1, T 2 and T 3 are stable causal operators, that is, T 1, T 2 and T 3 B c (l 2, l 2 ). The Youla parameter Q := K(I + P K) 1 is then an operator belonging to B c (l 2, l 2 ), and is related univoquely to the controller K. Note that Q is allowed to be time-varying. The magnitude of the signals w and z is measured in the l 2 - norm. The problem considered here is disturbance rejection. In this case the performance index can be written in the following form µ := inf { T zw : K being stabilizing linear time varying controller} = inf Q B c (l 2,l 2 ) T 1 T 2 QT 3 (4) The performance index (4) will be transformed into a distance minimization between a certain operator and a subspace to be specified shortly. To this end, define a nest N as a family of closed subspaces of the Hilbert space l 2 containing {0} and l 2 which is closed under intersection and closed span. Let Q n := I P n, for n = 1, 0, 1,, where P 1 := 0 and P := I. Then Q n is a projection, and we can associate to it the following nest N := {Q n l 2, n = 1, 0, 1, } (5) The triangular or nest algebra T (N ) is the set of all operators T such that T N N for every element N in N. That is T (N ) = {A B(l 2, l 2 ) : P n A(I P n ) = 0, n} = {A B(l 2, l 2 ) : (I Q n)aq n = 0, n} (6) Note that the Banach space B c (l 2, l 2 ) of causal and bounded operators may be viewed as a nest algebra since causal operators leave N invariant, i.e., for all operators A B c (l 2, l 2 ), AQ n l 2 Q n l 2 for all n. In fact B c (l 2, l 2 ) is exactly T (N ) the algebra of lower triangular operators. For N belonging to the nest N, N has the form Q n l 2 for some n. Define N = {N N : N < N} (7) N + = {N N : N > N} (8) where N < N means N N, and N > N means N N. The subspaces N N are called the atoms of N. Since in our case the atoms of N span l 2, then N is said to be atomic [3]. By analogy with the standard H theory [21], [22], and following [1], [3], [13] we introduce inner-outer factorizations for the operators in T (N ) as follows: An operator A in T (N ) is called outer if the range projection P (R A ), R A being the range of A and P the orthogonal projection onto R A, commutes with N and AN is dense in N R A for every N N. A partial isometry U is called inner in T (N ) if U U commutes with N

3 [1], [3], [13]. In our case, A T (N ) = B c (l 2, l 2 ) is outer if P commutes with each Q n and AQ n l 2 is dense in Q n l 2 Al 2. U B c (l 2, l 2 ) is inner if U is a partial isometry and U U commutes with every Q n. Applying these notions to the time-varying operator T 2 B c (l 2, l 2 ) we get T 2 = T 2i T 2o, where T 2i and T 2o are inner outer operators in B c (l 2, l 2 ), respectively. Similarly, co-inner-coouter fatorization can be defined, and the operator T 3 can be factored as T 3 = T 3co T 3ci, where T 3ci B c (l 2, l 2 ) is co-inner that is T 3ci is inner, T 3co B c (l 2, l 2 ) is co-outer, that is, T 3co is outer. The performance index µ in (4) can then be written as µ = inf Q B c(l 2,l 2 ) T 1 T 2i T 2o QT 3co T 3ci (9) Following the classical H control theory [21], [22], [24], [23], we assume (A1) that T 2o and T 3co are invertible both in B c (l 2, l 2 ). This assumption guarantees the bijection of the map Q T 2o B c (l 2, l 2 )T 3co In the time-invariant case this assumption means essentially that the outer factor of the plant P is invertible [22], [24]. Under this assumption T 2i becomes an isometry and T 3ci a co-isometry in which case T 2i T 2i = I and T 3ci T 3ci = I. By absorbing the operators T 2o and T 3co into the free operator Q, expression (9) is then equivalent to µ = inf Q B c (l 2,l 2 ) T 2iT 1 T 3ci Q (10) Expression (9) is the distance from the operator T 2i T 1T 3ci B(l 2, l 2 ) to the nest algebra B c (l 2, l 2 ). In the next section we study the distance minimization probblem (9) in the context of the operator algebra setting discussed above. III. EXISTENCE OF A PREDUAL AND AN OPTIMAL CONTROLLER Let X ba a Banach space and X its dual space, i.e., the space of bounded linear functionals defined on X. For a subset J of X, the annihilator of J in X is denoted J and is defined by [10] J := {Φ X : Φ(f) = 0, f J} (11) Similarly, if K is a subset of X then the preannihilator of K in X is denoted K, and is defined by K := {x X : Φ(x) = 0, Φ K} (12) The existence of a preannihilator implies that the following identity holds [10] min x y = sup < x, k > (13) y K k K, k 1 where <, > denotes the duality product. Let us apply these results to the problem given in (10) by letting X = B(l 2, l 2 ) x = T 2iT 1 T 3ci B(l 2, l 2 ) J = B c (l 2, l 2 ) Introduce a class of compact operators on l 2 called the trace-class or Schatten 1-class, denoted C 1, under the traceclass norm [17], [3], T 1 := tr(t T ) 1 2 (14) where tr denotes the Trace. We identify B(l 2, l 2 ) with the dual space of C 1, C 1, under the trace duality [17], that is, every A in B(l 2, l 2 ) induces a continuous linear functional on C 1 as follows: Φ A in C 1 is defined by Φ A (T ) = tr(at ), and we write B(l 2, l 2 ) C 1 To compute the preannihilator of B c (l 2, subspace S of C 1 by l 2 ) define the S := {T C 1 : (I Q n )T Q n+1 = 0, for all n} (15) In the following Lemma we show that S is the preannihilator of B c (l 2, l 2 ). Lemma 1: The preannihilator of B c (l 2, l 2 ) in C 1, B c (l 2, l 2 ), is isometrically isomorphic to S. Proof. By Lemma 16.2 in [3] the preannihilator of T (N ) is given by Φ T B c (l 2, l 2 ) if and only T belongs to the subspace {T C 1 : P (N ) T P (N) = 0 for all N in N } where P (N) denotes the orthogonal projection on N, likewise for P (N ). P (N ) the complementary projection of P (N ), that is, P (N ) = I P (N ). In our case N is atomic, and for any N N there exists n such that N = Q n+1 l 2, i.e., P (N) = Q n+1. The immediate predecessor of N, N, is then given by Q n l 2, i.e., N = Q n l 2. The orthogonal complement of N is then l 2 Q n l 2. So P (N ) = I Q n. Therefore, S = {T C 1 : (I Q n )T Q n+1 = 0, n} is isometrically isomorphic to the preannihilator of B c (l 2, l 2 ), and the Lemma is proved. The existence of a predual C 1 and a preannihilator S implies the following Theorem which is a consequence of Theorem 2 in [10] (Chapter 5.8). Theorem 1: Under assumption (A1) there exists an optimal Q o in B c (l 2, l 2 ) achieving optimal performance µ in (10), moreover the following identities hold µ = inf 2iT 1 T Q B c(l 2,l 2 3ci Q ) = T2iT 1 T3ci Q o = sup tr(t T2iT 1 T3ci) (16) T S, T 1 1

4 Theorem 1 not only shows the existence of an optimal LTV controller, but plays an important role in its computation by reducing the problem to a dual of finite dimensional convex optimizations. Under a certain condition in the next section it is shown that the supremum in (16) is achieved. Theorem 1 also leads to a solution based on a Hankel type operator, which parallel the Hankel operator know in the H control theory. IV. ALLPASS PROPERTY OF THE OPTIMUM: ALIGNMENT IN THE DUAL The supremum on the RHS of (16) may be achieved if the preannihilator S is enlarged. For this let us defined K as the space of compact operators acting on l 2 into l 2. The dual space of B(l 2, l 2 ) is then given by the space C 1 1 K [17], B(l 2, l 2 ) C 1 1 K (17) where K is the annihilator of K and the symbol 1 means that if Φ C 1 1 K then Φ has a unique decomposition as follows Φ = Φ o + Φ T (18) Φ = Φ o + Φ T (19) where Φ o K, and Φ T is induced by the operator T C 1. Banach space duality asserts that [10] inf x y = max Φ(x) (20) y J Φ J, Φ 1 In our case J = B c (l 2, l 2 ). Since B(l 2, l 2 ) contains C 1 as a subspace, then B c (l 2, l 2 ) contains the preannihilator S, i.e., the following expression may be deduced J := B c (l 2, l 2 ) = S 1 (K B c (l 2, l 2 )) (21) A deep result in [3] asserts that if a linear functional Φ belongs to the annihilator B c (l 2, l 2 ), and Φ decomposes as where Φ o and Φ T S. Then and as well. Φ = Φ o + Φ T ( ) K B c (l 2, l 2 ) Φ o B c (l 2, l 2 ) Φ T B c (l 2, l 2 ) Combining Theorem 1, (20) and (21) the following Lemma is deduced. If Lemma 2: Under assumption (A1) the following holds min T2iT 1 T Q B c(l 2, l 2 3ci Q = ) max Φ o (K B c ) T S, Φ o + T 1 1 Φ o (T 2iT 1 T 3ci) + tr(t T 2iT 1 T 3ci) (22) Φ opt = Φ opt,o + Φ Topt achieves the maximum in the RHS of (22), and Q o the minimum on the LHS, then the alignment condition in the dual is given by Φ opt,o (T 2iT 1 T 3ci) + tr(t opt T 2iT 1 T 3ci) = T 2iT 1 T 3ci Q o ( Φ opt,o + T opt 1 ) (23) If we further assume that (A2): T2i T 1T3ci is a compact operator. This is the case, for example, if anyone of T2i, T 1 or T3ci is compact, then Φ opt,o(t2i T 1T3ci ) = 0 and the maximum in (22) is achieved on S, that is the supremum in (16) becomes a maximum. It is instructive to note that in the linear time-invariant case assumption (A2) is the analogue of the assumption that T2i T 1T3ci is continuous on the unit circle, in which case the optimum is allpass [21], [24], [20]. In the linear time varying case allpass property of the optimum is formulated in the following Theorem. Theorem 2: Under assumptions (A1) and (A2) any optimal linear time varying Q o B c (l 2, l 2 ) satisfies the allpass condition T 2iT 1 T 3ci Q o = tr(t o T 2iT 1 T 3ci) (24) = max n (I Q n)t 2iT 1 T 3ciQ n (25) where T o is some operator in S, and T o 1 = 1. Proof. Identity (24) is implied by the previous argument that the supremum in (16) is achieved by some T o in S with trace-class norm equal to 1. Combining this result with Corollary 16.8 in [3] (see also [1]), which asserts that T2iT 1 T3ci Q o = sup (I Q n )T2iT 1 T3ciQ n n shows in fact that the supremum w.r.t. n is achieved proving that (25) holds. In fact, it can be shown that the operator T 2i T 1T 3ci Q o corresponds to an isometry. Identity (24) represents the allpass condition in the time-varying case, since it corresponds exactly to the allpass or flatness condition in the time-invariant case for the standard optimal H problem. In the next section, we relate our problem to an LTV operator analogous to the Hankel operator, which is known to solve the standard optimal H problem in the LTI case [21], [23].

5 V. A SOLUTION BASED ON A HANKEL OPERATOR The starting point here is the identity (16) µ = inf 2iT 1 T Q B c(l 2,l 2 3ci Q ) = sup A S, T 1 1 tr(at2it 1 T3ci) We need now to introduce the class of compact operators on l 2 called the Hilbert-Schmidt or Schatten 2-class, denoted C 2, under the Hilbert-Schmidt norm [17], [3], A 2 := ( tr(a A) ) 1 2 (26) In [27], [26] it is shown that any operator A in T (N ) C 1 admits a Riesz factorization, that is, there exist operators A 1 and A 2 in T (N ) C 2 such that A factorizes as A = A 1 A 2 (27) and A 1 = A 1 2 A 2 2 (28) A Hankel form [, ] B associated to a bounded linear operator B B(l 2, l 2 ) is defined by [27], [26] [A 1, A 2 ] B = tr(a 1 BA 2 ) (29) Since any operator in the preannihilator S belongs also to T (N ) C 1, then any A S factorizes as in (27). And if A 1 1, as on the LHS of (16), A S, the operators A 1 and A 2 both in T (N ) C 2 can be chosen such that A 1 2 1, A 2 2 1, and for all atoms n := Q n+1 Q n, n A 1 n = 0, n = 0, 1, 2, [26]. Define the orthogonal projection P of C 2 onto B c C 2, and write P + for the orthogonal projection with range the subspace of operators T in T (N ) C 2 such that n T n = 0, n = 0, 1, 2,. Introducing the notation (B 1, B 2 ) = tr(b2 B 1 ), and the Hankel form associated to B := T2i T 1T3ci, we have by a result in [26] [A 1, A 2 ] B = tr(ba 2 ) (30) = ( A 1, (T 2iT 1 T 3ci A 2 ) ) (31) = ( P + A 1, (BA 2 ) ) (32) = ( A 1, P + (BA 2 ) ) (33) = ( A 1, (H BA 2 ) ) (34) where H B is the Hankel operator (I P)BP, and we can see from above that its operator norm coincides with the norm of the Hankel form computed on A 1 and A 2 of unit Schatten 2-norm and satisfying the conditions above. Factorizing the operator T in (16) as T = t 1 t 2, where T 1 = t 1 2 t 2 2 = 1, t 1, t 2 T (N ) C 2, shows that following Theorem must hold. Theorem 3: µ = H T 2i T 1 T 3ci (35) = (I P)T 2iT 1 T 3ciP (36) Finally in the next section we show that computing µ, within desired tolerances, reduces to two finite dimensional convex optimizations in the entries of Q. VI. APPROXIMATION BY FINITE DIMENSIONAL CONVEX OPTIMIZATIONS If {e n : n = 0, 1, 2, } is the standard orthonormal basis in l 2, then Q n l 2 is the linear span of {e k : k = n + 1, n + 2, }. The matrix representation of A T (N ) w.r.t. this basis is lower triangular. Since P n = I Q n I as n in the strong operator topology (SOT), if we restrict the miminimization in (16) over Q B c = T (N ) to the span of {e n : n = 0, 1, 2,, N}, that is, P N l 2 =: l 2 N we get a finite dimensional convex optimization problem in lower triangular matrices Q N of dimension N, that is, µ N := inf Q N B c (P N l 2,P N l 2 ) (T 2iT 1 T 3ci) N Q N which overestimates µ and result in upper bounds. Solving such problems are then applications of convex programming techniques. However, this is not of much use unless we know how far from the optimum µ this approximation has caused us to go. Applying the same argument to the dual optimization on the RHS of (16), that is, where µ N := sup tr(t N (T2iT 1 T3ci) N T N S N, T N 1 1 S N := {T N C 1 (P N l 2, P N l 2 ) : (I Q n )T N Q n+1 = 0, for all n} yields other finite dimensional convex optimizations in T N as the restrictions of T to l 2 N. We get lower bounds for µ since the dual optimization involves a supremum rather than an infimum. Since P N I as N in the SOT, it can be shown that these upper and lower bounds converge to the optimum µ as N. These optimizations estimate µ within known tolerance and compute the corresponding LTV operators Q N, which in turn allow the computation of LTV controllers K through the Youla parametrization. REFERENCES [1] Arveson W. Interpolation problems in nest algebras, Journal of Functional Analysis, 4 (1975) [2] Chapellat H., Dahleh M. Analysis of time-varying control strategies for optimal disturbance rejection and robustness, IEEE Transactions on Automatic Control, 37 (1992) [3] Davidson K.R. Nest Algebras, Longman Scientific & Technical, UK, [4] Djouadi S.M. optimization of Highly Uncertain Feedback Systems in H, Ph.D. thesis, McGill University, Montreal, Canada, [5] Djouadi S.M. Optimal Robust Disturbance Attenuation for Continuous Time-Varying Systems, Proceedings of CDC, vol. 4, (1998) [6] Djouadi S.M. Optimal Robust Disturbance Attenuation for Continuous Time-Varying Systems, International journal of robust and non-linear control, vol. 13, (2003) [7] Douglas R.G. Banach Algebra Techniques in Operator Theory, Academic Press, NY, [8] Khammash M., Dahleh M. Time-varying control and the robust performance of systems with structured norm-bounded perturbations, Proceedings of the IEEE Conference on Decision and Control, Brighton, UK, 1991.

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